For expository convenience, the present invention will be illustrated with reference to one particular application thereof, namely as a group delay estimator in a network analyzer. However, it should be recognized that the invention is not so limited.
In the field of the electrical arts, it is often necessary to analyze the electrical characteristics of a configuration of interconnected electrical components or "network." One characteristic of networks that is often of interest is the length of time required for a signal to travel from an input to an output. A measurement made on networks and related to this characteristic is known as the group delay. The group delay of a network is defined as the negative of the derivative of the phase response of the network with respect to frequency or, in other words, the rate at which the phase response changes with frequency.
Traditionally, the group delay of a network has been determined by measuring the phase response of the network at two different frequencies using a stimulus signal applied to an input of the network. The difference between the two frequencies is termed the "aperture." The group delay of the network at a frequency in the center of the aperture is calculated as the difference in unwrapped or cumulative phase between the two phase response measurements divided by the aperture width. Usually, the phase difference is measured in radians and the aperture width is measured in radians per second to yield a group delay measurement in seconds. However, other units can be used.
Previously, group delay measurements required special purpose, dedicated hardware. More recently however, electronic measurement instruments known as network analyzers have been used. Network analyzers can be used to provide a trace of the phase response of a network at discrete frequency intervals. The group delay of the network is determined from the traces using the traditional method. Two points on the network analyzer's trace corresponding to the phase response of the network at two given frequencies are subtracted, then divided by the difference between the given frequencies. The result is an approximation of the derivative of the network's phase response as a function of frequency. The negative of the result is therefore an estimate of the group delay. The method can be repeated at discrete intervals along the phase response trace to generate a group delay trace.
The traditional method of group delay estimation generates two types of error from the true group delay that are dependent on the aperture width. First, noise or random measurement errors in the phase response measurements cause corresponding errors in the group delay estimates. The effect of the noise errors on the group delay estimates can be decreased by increasing the aperture width. For example, if the aperture width is doubled and the noise variance of the phase response measurements is constant, the variance of the group delay estimates is halved.
Larger aperture widths, however, have a smoothing effect on the group delay trace, resulting in a group delay trace which is distorted from the actual group delay. In places where the actual group delay is changing rapidly, for example, the group delay trace produced with a large aperture width is smoother than the actual group delay. This distortion from the actual group delay is a second type of error. So, large aperture widths are desirable for minimizing noise variance in the group delay trace, while small aperture widths are desirable for minimizing distortion of the trace. The selection of aperture size becomes a tradeoff between conflicting desired noise minimization characteristics to preserve the true shape of the group delay trace.
The present invention provides a more accurate measurement of the group delay which has a decreased noise variance over a sequence of frequencies as compared to the traditional method. The present invention also provides a decreased trace distortion compared to the traditional method. In accordance with a preferred embodiment of the invention, the phase response of a network is measured at a sequence of stimulus signal frequencies. Using the unwrapped or cumulative phase response at a number of these frequencies within a frequency aperture, the phase response rate of change as a function of frequency is determined using linear regression techniques. Since the phase response rate of change as a function of frequency is the group delay, the result of the linear regression analysis yields a group delay estimate at a frequency in the center of the aperture. This group delay estimate determination is repeated at a sequence of frequencies to create a trace of the network's group delay across a range of frequencies.
The method of the present invention is also applicable to estimating traces of measurements other than group delay that are the derivative of a measurement trace. For example, the method can be applied to estimate a trace of the instantaneous frequency of a signal over time from a trace of the signal's phase measured at discrete sample times. In a prior frequency estimation system, described in U.S. Pat. No. 4,983,906 to Hiller, a single frequency estimate is calculated using linear regression analysis techniques. The frequency estimate calculated by the Hiller system assumes that the frequency of the signal remains constant over time. However, it is often useful to monitor signals where the instantaneous frequency changes over time such as phase or frequency modulated signals or signals with drifting frequencies. With the present invention, it is possible to form a trace estimate of the instantaneous frequency over time.